# Modeling Temperature Dependence in Marangoni-driven Thin Films

A thin liquid film is a layer of fluid that has a breadth much greater than its depth.  A representative example is the tear film that coats your eye to protect it: the thickness of this coating is only a few micrometers but its breadth is a few centimeters, so the aspect ratio between the two length scales is about $10^{-4}$.  It is difficult to control such thin films because direct contact with them can lead to rupture.By studying how physical properties,such as surface tension, affect the flow of thin films they can, however, be controlled indirectly.

Temperature gradients and surfactants are the two primary ways in which surface tension gradients, which generate what are called Marangoni driving stresses, can be produced in order to affect the flow of thin liquid films. My dissertation looks at a plate-coating application in which thermal Marangoni stresses serve as a means of controlling the thickness of the final coating.  This work entails reducing the full Navier-Stokes equations with 3 spatial variables down to a fourth-order nonlinear partial differential equation boundary-value problem with only 1 spatial variable through the use of nondimensionalization, scaling, and asymptotic analysis, making using of the small aspect ratio in what is called the Reynolds lubrication approximation.

In order to study the mathematical model obtained, a boundary-value problem is solved numerically. This problem is used to study system behavior in a variety of parameter regimes in order to better understand transitions between qualitatively distinct solution classes.  One goal of this characterization of solution classes is to determine how best to obtain a thin film coating of a desired thickness by adjusting the control parameters.

Harrison David Parke Potter graduated in May 2016 with a PhD in Mathemaics. His thesis work, advised by Professor Thomas P. Witelski, is deiscribed above.  After his PhD, Harrison Potter became an Assistant Professor of Mathematics in the Department of Mathematics, Computing, and Information Systems at Marietta College.